Strong solutions to the 3D primitive equations with only horizontal dissipation: Near H1 initial data

In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data (v0,T0)∈H1, by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data (v0,T0)∈H1∩L∞, such that ∂zv0∈Lm, for some m∈(2,∞), by using the logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality. This paper improves the previous results obtained in Cao et al. (2016) [10], where the initial data (v0,T0) was assumed to have H2 regularity.